Properties

Label 2-966-161.97-c1-0-28
Degree $2$
Conductor $966$
Sign $-0.888 + 0.459i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.755 + 0.654i)3-s + (0.841 + 0.540i)4-s + (−0.486 + 3.38i)5-s + (−0.540 − 0.841i)6-s + (0.277 − 2.63i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (1.42 − 3.10i)10-s + (−1.18 − 4.02i)11-s + (0.281 + 0.959i)12-s + (−5.10 − 2.33i)13-s + (−1.00 + 2.44i)14-s + (−2.58 + 2.23i)15-s + (0.415 + 0.909i)16-s + (−4.84 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (0.436 + 0.378i)3-s + (0.420 + 0.270i)4-s + (−0.217 + 1.51i)5-s + (−0.220 − 0.343i)6-s + (0.105 − 0.994i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (0.449 − 0.983i)10-s + (−0.356 − 1.21i)11-s + (0.0813 + 0.276i)12-s + (−1.41 − 0.646i)13-s + (−0.269 + 0.653i)14-s + (−0.667 + 0.578i)15-s + (0.103 + 0.227i)16-s + (−1.17 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.888 + 0.459i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.888 + 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00550739 - 0.0226559i\)
\(L(\frac12)\) \(\approx\) \(0.00550739 - 0.0226559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
3 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (-0.277 + 2.63i)T \)
23 \( 1 + (-1.32 - 4.61i)T \)
good5 \( 1 + (0.486 - 3.38i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (1.18 + 4.02i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (5.10 + 2.33i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (4.84 - 3.11i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (2.05 + 1.32i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (2.02 - 1.30i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (3.22 - 2.79i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + (-6.12 + 0.880i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (5.88 + 0.846i)T + (39.3 + 11.5i)T^{2} \)
43 \( 1 + (8.91 + 7.72i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + (7.56 - 3.45i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-11.9 - 5.45i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-1.76 - 2.03i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-2.40 + 8.19i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-4.91 - 1.44i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (6.49 - 10.1i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (5.00 + 2.28i)T + (51.7 + 59.7i)T^{2} \)
83 \( 1 + (-1.67 - 11.6i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (1.92 - 2.22i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-0.0367 + 0.255i)T + (-93.0 - 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.995162147665427037821767451474, −8.793492260040821732918574291928, −8.006308808489890297611037541121, −7.19331231258471521016375859008, −6.69227159239494873952813830778, −5.31127317830312886618338026193, −3.85274969216185482212927419598, −3.18037826871302982111161383313, −2.19513485978657747365071599030, −0.01156876228823277145661234970, 1.81241896725448696210564925850, 2.50096597423314071277081427850, 4.63602071618772016351142968213, 4.87461053569541186830026078552, 6.29033398123194738783105762385, 7.23375763069214154637007612996, 8.054569881750889581344903975700, 8.705883332792943817066955388049, 9.462032302731885844816216993680, 9.783015093189487005282069989234

Graph of the $Z$-function along the critical line